Contrast this with the levitical "Put to death" (say Lev 20:12 for example).
The use of "murder" in place of "kill" in Ex 20:13 and Mt 5:21-22 would be a translation far more consistent with the rest of the bible (and still holds true to the original words)
I believe your interpretation is correct as with Ex 21:12. "Kill" in the first instance of Ex 20:13 meaning an action, rather than as a punishment for the act itself, i. e., "put to death." Does that make sense to you.